The word `local' is important here. In absence of the above mentioned motivation stemming from the space-time physics, the quantum Carrollian boost generator in [1]} was given non-zero diagonal entries in matrix representations in the Hilbert space. It was shown in [2]} that the correlation function, first derived in [3]}, between two fields corresponding to states (by the assumption of state-operator correspondence) transforming under such a `singlet' representation (as defined in [1]}) can arise from a non-local action. The conclusion that this correlator can originate only from a non-local action can be reached by taking a Fourier transformation of the same: the momentum-space form of the Carrollian correlator makes it manifest that the equation of motion, of which this correlator is a Green's function, must contain spatial derivatives of negative order whenever \(\xi \ne 0\) , irrespective of \(\Delta \)  (notations are as used in [1]} whose \(\xi =0\)  counterpart is [6]}). This conclusion also applies for any higher rank multiplets [1]} with \(\xi \ne 0\) . Moreover, the `limiting' perspective in [8]} has given rise only to such `singlet' fields. Instead, in this paper, we will be concerned with exploring the properties of CCA\(_{1+1}\)  invariant field theories describable by local actions.
